Robust Bayesianism: Imprecise and Paradoxical Reasoning

We are interested in understanding the relationship between Bayesian inference and evidence theory, in particular imprecise and paradoxical reasoning. The concept of a set of probability distributions is central both in robust Bayesian analysis and in some versions of Dempster-Shafer theory. Most of the literature regards these two theories as incomparable. We interpret imprecise probabilities as imprecise posteriors obtainable from imprecise likelihoods and priors, both of which can be considered as evidence and represented with, e.g., DS-structures. The natural and simple robust combination operator makes all pairwise combinations of elements from the two sets. The DS-structures can represent one particular family of imprecise distributions, Choquet capacities. These are not closed under our combination rule, but can be made so by rounding. The proposed combination operator is unique, and has interesting normative and factual properties. We compare its behavior on Zadeh’s example with other proposed fusion rules. We also show how the paradoxical reasoning method appears in the robust framework.